Analytical Swarm Chemistry: Characterization and Analysis of Emergent Swarm Behaviors

TL;DR

This work introduces Analytical Swarm Chemistry to address the unpredictability of emergent swarm behaviors by linking microstate observations to macrostates through information markers and phase diagrams. By mapping parameters such as $N$, $v$, $\omega$, $\gamma$, and $\phi$ to emergent macrostates $B_j$, it provides a principled way to identify parameter regions where behaviors reliably arise, rather than pursuing a single optimal point. Case studies on milling and diffusion with minimal binary controllers demonstrate sufficient conditions for these macrostates, and real-world robot validation with TurboPis and RSRS corroborates key predictions from the simulations. The framework thus offers a interpretable path toward predictable, scalable swarm deployment and points to future work in heterogeneous swarms and automated parameter-space exploration.

Abstract

Swarm robotics has potential for a wide variety of applications, but real-world deployments remain rare due to the difficulty of predicting emergent behaviors arising from simple local interactions. Traditional engineering approaches design controllers to achieve desired macroscopic outcomes under idealized conditions, while agent-based and artificial life studies explore emergent phenomena in a bottom-up, exploratory manner. In this work, we introduce Analytical Swarm Chemistry, a framework that integrates concepts from engineering, agent-based and artificial life research, and chemistry. This framework combines macrostate definitions with phase diagram analysis to systematically explore how swarm parameters influence emergent behavior. Inspired by concepts from chemistry, the framework treats parameters like thermodynamic variables, enabling visualization of regions in parameter space that give rise to specific behaviors. Applying this framework to agents with minimally viable capabilities, we identify sufficient conditions for behaviors such as milling and diffusion and uncover regions of the parameter space that reliably produce these behaviors. Preliminary validation on real robots demonstrates that these regions correspond to observable behaviors in practice. By providing a principled, interpretable approach, this framework lays the groundwork for predictable and reliable emergent behavior in real-world swarm systems.

Figures (8)

• Figure 1: (a) Abstract phase diagram showing where $B_j = 1$, depicted as the green area, in respect to Parameter A and B. (b) Abstract phase diagram that uses intensity of color to represent how frequently behavior $B_j$ occurred out of the total runs simulated.
• Figure 2: Examples of local interaction rules and their resulting emergent behaviors. The top row shows the controller \ref{['eq:mill_control']} and the corresponding milling behavior it can produce, while the bottom row shows the controller \ref{['eq:diff_controller']} and its resulting diffusion behavior.
• Figure 3: Examples of different circliness values ($Y_2 = \overline{c}$) used to define the milling behavior. If we assume that the average speed is equal to the set speed of the system $v$ (i.e. the agents are moving constantly), then only the top-left snapshot would be considered milling, as $\overline{c} = 0.004$ therefore $M^1 \in \eta^1$, satisfying the criteria for this behavior.
• Figure 4: Phase diagrams of number of agents $N$ vs FOV angle $\phi$ showing: (a) average speed $Y_1 = \bar{v}$, (b) circliness values $Y_2 = \bar{c}$, (c) $B_{1}$ value. (a) and (b) display some important region in circliness and average speed, respectively, but it is made even more clear when both are considered together and compared to $\eta^{mill}$ to identify when milling is occurring.
• Figure 5: Examples of different nearest-neighbor variance values ($Y_3 = \overline{\delta}$) used to define the diffusion behavior. Only the top-left snapshot is considered diffusing, as $\overline{\delta} = 0.001 \in \eta^2$, satisfying the criteria for this information marker.